These problems are complex fluid mechanics scenarios that can be solved using principles like Bernoulli's equation, Torricelli's law, and the conservation of energy. I'll go through the solutions step-by-step.

Problem 1: Discharge from a Tank with Acceleration

Given:

• Diameter of the orifice, $d = 100 \, \text{mm} = 0.1 \, \text{m}$
• Depth above the orifice, $h = 9 \, \text{m}$
• Acceleration downward $a = \frac{g}{2}$
• Coefficient of discharge $C_d = 0.64$
• Coefficient of velocity $C_v = 0.98$

The effective acceleration due to gravity:

$g' = g - a = g - \frac{g}{2} = \frac{g}{2} = 4.905 \, \text{m/s}^2$

The theoretical velocity of the jet is:

$v = \sqrt{2gh} = \sqrt{2 \times 4.905 \times 9} \approx 9.39 \, \text{m/s}$

Using the coefficient of velocity $C_v$:

$v_{\text{actual}} = C_v \times v = 0.98 \times 9.39 \approx 9.20 \, \text{m/s}$

The discharge $Q$ is given by:

$Q = C_d \times A \times v_{\text{actual}}$

Where $A$ is the area of the orifice:

$A = \frac{\pi d^2}{4} = \frac{\pi (0.1)^2}{4} = 7.854 \times 10^{-3} \, \text{m}^2$

So,

$Q = 0.64 \times 7.854 \times 10^{-3} \times 9.20 \approx 0.0463 \, \text{m}^3/\text{s}$

Thus, the discharge is $0.0463 \, \text{m}^3/\text{s}$.

Problem 2: Water Discharge Over Time

To solve this problem, we first need to determine the initial and final horizontal ranges and velocities of the water jet using kinematic equations. Then, we can calculate the volume of water discharged.

Problem 3: Orifice Area in a Canal Lock

Given:

• Width of the canal lock: $W = 6 \, \text{m}$
• Length of the canal lock: $L = 24 \, \text{m}$
• Time to empty: $t = 5 \, \text{minutes} = 300 \, \text{seconds}$
• Head: $H = 5.5 \, \text{m}$
• Discharge coefficient: $C = 0.60$

This problem involves calculating the area using the flow rate equation:

$Q = \frac{V}{t} \quad \text{and} \quad Q = C \times A \times \sqrt{2gH}$

Where $V$ is the volume of the lock.

Problem 4: Time Interval for Change in Jet Velocity

Given:

• Diameter of the tank: 3 m
• Orifice diameter: 0.1 m
• Initial head: 1 m
• Initial and final velocities: 10 m/s to 7 m/s

This problem requires the application of Torricelli’s law and an integration approach to find the time interval.

Problem 5: Jet Issued from a Tank

The jet's maximum height can be found using projectile motion principles, while the horizontal distance (X) and time are determined using kinematics.

Do you need a detailed solution for any of the above problems?

Related Questions:

1. How does the acceleration of the tank affect the discharge in fluid dynamics problems?
2. What are the key differences between the coefficients of discharge, velocity, and contraction in fluid flow through orifices?
3. How do you calculate the horizontal range of a water jet from an elevated tank?
4. What is the importance of air resistance in real-world fluid dynamics applications?
5. How can you determine the time for fluid levels to change in a cylindrical tank with an orifice?

Tip:

Always double-check the units when calculating fluid discharge, especially when dealing with metric conversions.